11 and 15

2. ## Re: class 10 question

For 11, you are trying to show that $n^2+n+1 \equiv 0 \pmod{5}$ is false for all $n$. This breaks down to five cases ($n \equiv 0 \pmod{5}, n\equiv 1 \pmod{5}, n\equiv 2 \pmod{5}, n\equiv 3\pmod 5, n\equiv 4\pmod 5$). Plug in those five values for $n$, and show that in no case do you get congruence with zero.

For 15, replace $n$ with $2k+1$. Multiply out. You get $4k(k+1)$. For any integer $k$, either $k$ or $k+1$ is even. Thus, you either have $k=2r$ or $k+1=2r$ for some integer $r$. This gives either $8r(k+1)$ or $8kr$. In either case, it is divisible by 8.

3. ## Re: class 10 question

When you opened an account on this forum, you were asked to state that you had read the forum rules. Had you actually done that, you would have seen that "rule 5" is
"(5) When you post a question include any work that you have been able to do and show us
where you are stuck. "

You have now posted two questions in which you have not shown any work or any indication that you understand what the question is giving you or what the question is asking. If you really do not understand the basics then you need to speak to your instructor.